1 Foraging Model
- 1.1 Dive profiles per indidivuals
- 1.2 Data Statistics before track cut
2 Data Cleaning Rules.
3 Temporal Variation in Dive Behavior
4 Raw Argos Data
5 Comparison with 2d movement model
- 5.1 Chains
- 5.2 Posterior Checks

1 Foraging Model

The proportion of time an animal is in a feeding behavioral state.

Process Model

\[Y_{i,t+1} \sim Multivariate Normal(d_{i,t},σ)\]

\[d_{i,t}= Y_{i,t} + γ_{s_{i,g,t}}*T_{i,g,t}*( Y_{i,g,t}- Y_{i,g,t-1} )\]

\[ \begin{matrix} \alpha_{i,1,1} & \beta_{i,1,1} & 1-(\alpha_{i,1,1} + \beta_{i,1,1}) \\ \alpha_{i,2,1} & \beta_{i,2,1} & 1-(\alpha_{i,2,1} + \beta_{i,2,1}) \\ \alpha_{i,3,1} & \beta_{i,3,1} & 1-(\alpha_{i,3,1} + \beta_{i,3,1}) \\ \end{matrix} \] \[logit(\phi_{Behavior}) = \alpha_{Behavior_{t-1}} \] The behavior at time t of individual i on track g is a discrete draw. \[S_{i,g,t} \sim Cat(\phi_{traveling},\phi_{foraging},\phi_{resting})\]

Dive information is a mixture model based on behavior (S)

\(\text{Average dive depth}(\psi)\) \[ DiveDepth \sim Normal(dive_{\mu_S},dive_{\tau_S})\] \[ DiveDuration \sim Normal(duration_{\mu_S},duration_{\tau_S})\]

1.1 Dive profiles per indidivuals

Dive Profiles with Argos timestamps

1.2 Data Statistics before track cut

## # A tibble: 11 x 4
##    Animal     n argos  dive
##     <int> <int> <int> <int>
##  1 131111   548   173   375
##  2 131115  1029   179   850
##  3 131116  1932   457  1475
##  4 131127  9275  2495  6780
##  5 131128   112    52    60
##  6 131130  1165   151  1014
##  7 131132  2292   589  1703
##  8 131133  8497  2078  6419
##  9 131134  2098   653  1445
## 10 131136  6844  1970  4874
## 11 154187  1866   486  1380

## [1] 35658

2 Data Cleaning Rules.

All tracks start with an observed argos location.
A single track must have one argos location per time step.

2.1 Data statistics after track cut

## # A tibble: 11 x 2
##    Animal     n
##     <int> <int>
##  1 131111   454
##  2 131115  1020
##  3 131116  1918
##  4 131127  9047
##  5 131128   106
##  6 131130   511
##  7 131132  1713
##  8 131133  8193
##  9 131134  2018
## 10 131136  6721
## 11 154187  1847

## # A tibble: 11 x 2
##    Animal Tracks
##     <int>  <int>
##  1 131111      1
##  2 131115      1
##  3 131116      2
##  4 131127     11
##  5 131128      1
##  6 131130      1
##  7 131132      1
##  8 131133      4
##  9 131134      1
## 10 131136      3
## 11 154187      1

2.2 Dive Prior Shapes

2.2.1 Dive Depth

sink(“Bayesian/ThreeState.jags”) cat(" model{

#from jonsen 2016

pi <- 3.141592653589

#for each if 6 argos class observation error

for(x in 1:6){

##argos observation error##
argos_prec[x,1:2,1:2] <- argos_cov[x,,]

#Constructing the covariance matrix
argos_cov[x,1,1] <- argos_sigma[x]
argos_cov[x,1,2] <- 0
argos_cov[x,2,1] <- 0
argos_cov[x,2,2] <- argos_alpha[x]
}

for(i in 1:ind){
for(g in 1:tracks[i]){

## Priors for first true location
#for lat long
y[i,g,1,1:2] ~ dmnorm(argos[i,g,1,1,1:2],argos_prec[1,1:2,1:2])

#First movement - random walk.
y[i,g,2,1:2] ~ dmnorm(y[i,g,1,1:2],iSigma)

###First Behavioral State###
state[i,g,1] ~ dcat(firstmove[]) ## assign state for first obs

#Process Model for movement
for(t in 2:(steps[i,g]-1)){

#Behavioral State at time T
phi[i,g,t,1] <- Traveling[state[i,g,t-1]] 
logit(phi[i,g,t,3]) <- alpha_hours[state[i,g,t-1]] + beta_hours[state[i,g,t-1]]* cos((2*pi*hours[i,g,t])/(24)) + beta2_hours[state[i,g,t-1]] * sin((2*pi*hours[i,g,t])/24)^2
phi[i,g,t,2] <- 1-(phi[i,g,t,1] + phi[i,g,t,3])

state[i,g,t] ~ dcat(phi[i,g,t,])

#Turning covariate
#Transition Matrix for turning angles
T[i,g,t,1,1] <- cos(theta[state[i,g,t]])
T[i,g,t,1,2] <- (-sin(theta[state[i,g,t]]))
T[i,g,t,2,1] <- sin(theta[state[i,g,t]])
T[i,g,t,2,2] <- cos(theta[state[i,g,t]])

#Correlation in movement change
d[i,g,t,1:2] <- y[i,g,t,] + gamma[state[i,g,t]] * T[i,g,t,,] %*% (y[i,g,t,1:2] - y[i,g,t-1,1:2])

#Gaussian Displacement in location
y[i,g,t+1,1:2] ~ dmnorm(d[i,g,t,1:2],iSigma)

}

#Final behavior state
phi[i,g,steps[i,g],1] <- Traveling[state[i,g,steps[i,g]-1]] 
logit(phi[i,g,steps[i,g],3]) <- alpha_hours[state[i,g,steps[i,g]-1]] + beta_hours[state[i,g,steps[i,g]-1]]* cos((2*pi*hours[i,g,steps[i,g]])/(24)) + beta2_hours[state[i,g,steps[i,g]-1]] * sin((2*pi*hours[i,g,steps[i,g]])/24)^2
phi[i,g,steps[i,g],2] <- 1-(phi[i,g,steps[i,g],1] + phi[i,g,steps[i,g],3])
state[i,g,steps[i,g]] ~ dcat(phi[i,g,steps[i,g],])

##  Measurement equation - irregular observations
# loops over regular time intervals (t)    

for(t in 2:steps[i,g]){

# loops over observed locations within interval t
for(u in 1:idx[i,g,t]){ 
zhat[i,g,t,u,1:2] <- (1-j[i,g,t,u]) * y[i,g,t-1,1:2] + j[i,g,t,u] * y[i,g,t,1:2]

#for each lat and long
#observed position
argos[i,g,t,u,1:2] ~ dmnorm(zhat[i,g,t,u,1:2],argos_prec[argos_class[i,g,t,u],1:2,1:2])

#for each dive depth
#dive depth at time t
dive[i,g,t,u] ~ dnorm(depth_mu[state[i,g,t]],depth_tau[state[i,g,t]])T(0,)

#Assess Model Fit

#Fit dive discrepancy statistics
eval[i,g,t,u] ~ dnorm(depth_mu[state[i,g,t]],depth_tau[state[i,g,t]])T(0,)
E[i,g,t,u]<-pow((dive[i,g,t,u]-eval[i,g,t,u]),2)/(eval[i,g,t,u])

dive_new[i,g,t,u] ~ dnorm(depth_mu[state[i,g,t]],depth_tau[state[i,g,t]])T(0,)
Enew[i,g,t,u]<-pow((dive_new[i,g,t,u]-eval[i,g,t,u]),2)/(eval[i,g,t,u])

}
}
}
}

###Priors###

#Process Variance
iSigma ~ dwish(R,2)
Sigma <- inverse(iSigma)

##Mean Angle
tmp[1] ~ dbeta(10, 10)
tmp[2] ~ dbeta(10, 10)
tmp[3] ~ dbeta(10, 10)

# prior for theta in 'traveling state'
theta[1] <- (2 * tmp[1] - 1) * pi

# prior for theta in 'foraging state'    
theta[2] <- (tmp[2] * pi * 2)

theta[3] <- (tmp[3] * pi * 2)

##Move persistance
# prior for gamma (autocorrelation parameter)

##Behavioral States

#gamma[1] ~ dbeta(3,2)
#dev ~ dunif(0,0.5)         ## a random deviate to ensure that gamma[1] > gamma[2]
#gamma[2] <- gamma[1] * dev ## 2d movement for foraging state
#dev2 ~ dunif(0,0.5)            ## a random deviate to ensure that gamma[1] > gamma[3]
#gamma[3] <- gamma[1] * dev2  ## 2d movement for resting state

gamma[1]<-0.7
gamma[2]<-0.3
gamma[3]<-0.3

#Temporal autocorrelation in behavior - remain in current state
Traveling[1] ~ dbeta(1,1)
Traveling[2] ~ dbeta(1,1)
Traveling[3] ~ dbeta(1,1)

#Temporal autocorrelation in behavior - transition Foraging
alpha_hours[1] ~ dnorm(0,0.386)
alpha_hours[2] ~ dnorm(0,0.386)
alpha_hours[3] ~ dnorm(0,0.386)

#Effect of time of day on transitioning to resting
beta_hours[1] <- 0 
beta_hours[2] ~ dnorm(0,0.386)
beta_hours[3] ~ dnorm(0,0.386)

#Transition from travel to resting
beta2_hours[1] <- 0
beta2_hours[2] ~ dnorm(0,0.386)
beta2_hours[3] ~ dnorm(0,0.386)

#Probability of initial behavior
firstmove ~ ddirch(rep(1,3))

#Foraging dives are deepest
depth_mu[2] <- 0.150
depth_mu[1] <- 0.02
depth_mu[3] <- 0.015

#depth and duration variance
depth_tau[1] <-2000
depth_tau[2] <- 300
depth_tau[3] <- 5000

##Observation Model##
##Argos priors##
#longitudinal argos precision, from Jonsen 2005, 2016, represented as precision not sd

#by argos class
argos_sigma[1] <- 11.9016
argos_sigma[2] <- 10.2775
argos_sigma[3] <- 1.228984
argos_sigma[4] <- 2.162593
argos_sigma[5] <- 3.885832
argos_sigma[6] <- 0.0565539

#latitidunal argos precision, from Jonsen 2005, 2016
argos_alpha[1] <- 67.12537
argos_alpha[2] <- 14.73474
argos_alpha[3] <- 4.718973
argos_alpha[4] <- 0.3872023
argos_alpha[5] <- 3.836444
argos_alpha[6] <- 0.1081156

}"
,fill=TRUE)

sink()

##     user   system  elapsed 
## 2464.286  571.552 9275.116

2.3 Chains

2.4 Temporal autocorrelation in foraging

Reconstructing the transition matrix

3 Temporal Variation in Dive Behavior

3.1 Simulate dive depths

3.2 Diel

3.3 Month

4 Raw Argos Data

5 Comparison with 2d movement model

Movement model

##    user  system elapsed 
##   0.573   0.131 459.074

5.1 Chains

Where does the 3d predict foraging that the 2d misses?

Where does the 3d predict resting that the 2d misses?

Where does the 2d predict foraging that the 3d refines to traveling?

Where does the 2d predict foraging that the 3d refines to resting?

5.2 Posterior Checks

The goodness of fit is a measured as chi-squared. The expected value is compared to the observed value of the actual data. In addition, a replicate dataset is generated from the posterior predicted intensity. Better fitting models will have lower discrepancy values and be Better fitting models are smaller values and closer to the 1:1 line. A perfect model would be 0 discrepancy. This is unrealsitic given the stochasticity in the sampling processes. Rather, its better to focus on relative discrepancy. In addition, a model with 0 discrepancy would likely be seriously overfit and have little to no predictive power.

## # A tibble: 1 x 2
##   `mean(E)` `var(Enew)`
##       <dbl>       <dbl>
## 1 0.1259669 0.008264216

Foraging Model

Ben Weinstein

2017-11-17 19:20:20